On the decomposition into Discrete, type II and type III $C^*$-algebras

TL;DR

This paper introduces a decomposition scheme for C*-algebras based on classifying them into discrete, type II, and type III categories, extending known decompositions and analyzing their structural properties.

Contribution

It develops a new framework for classifying C*-algebras into five types, showing these classes are stable under key algebraic operations and extending decomposition results from W*-algebras.

Findings

Existence of largest ideals for each type within any C*-algebra.

Decomposition of C*-algebras into continuous fields over a Hausdorff space.

Extension of W*-algebra decomposition to C*-algebras.

Abstract

We obtained a "decomposition scheme" of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsido), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to "classify" C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking "essential extension" and "normal quotient". Furthermore, there exist the largest discrete finite ideal $A_{d,1}$, the largest discrete essentially infinite ideal $A_{d,\infty}$, the largest type II finite ideal $A_{II,1}$, the largest type II essentially infinite ideal $A_{II,\infty}$, and the largest type III ideal $A_{III}$ of any C*-algebra $A$ such that $A_{d,1} + A_{d,\infty} + A_{II,1} + A_{II,\infty} + A_{III}$ is an essential ideal of $A$. This "decomposition" extends the corresponding one for $W^*$-algebras. We also give a closer look at C*-algebras with Hausdorff primitive spectrum, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above.